(a) Derive the polynomial of degree five that satisfies both the Legendre equation ( 1 − x 2 ) y ″ − 2 x y ′ + 30 y = 0 and the normalization condition y ( 1 ) = 1 . (b) Sketch your solution from (a) and determine approximations to all zeros and local maxima and local minima on the interval ( − 1 , 1 ) .
(a) Derive the polynomial of degree five that satisfies both the Legendre equation ( 1 − x 2 ) y ″ − 2 x y ′ + 30 y = 0 and the normalization condition y ( 1 ) = 1 . (b) Sketch your solution from (a) and determine approximations to all zeros and local maxima and local minima on the interval ( − 1 , 1 ) .
Solution Summary: The author explains how to find the polynomial of degree five that satisfies the Legendre equation.
(a) Derive the polynomial of degree five that satisfies both the Legendre equation
(
1
−
x
2
)
y
″
−
2
x
y
′
+
30
y
=
0
and the normalization condition
y
(
1
)
=
1
.
(b) Sketch your solution from (a) and determine approximations to all zeros and local maxima and local minima on the interval
(
−
1
,
1
)
.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
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